High School Physics Change Due to Heat

You might have heard off-hand that things expand when heated, and this is true. In fact, things expand a considerable amount when heated; nothing monumental like a ball tripling in size, but considerable nonetheless. We call this expansion a linear expansion, and as you’ll see this expansion is best shown in gases. This is because of the ideal gas law, which is the cause of so many amazing things that utilize heat exchanges. This video serves as kind of an example of what happens when temperature increases, and the next one will serve as a rule book and a reason why this happens.

I have a question, in example 4 when you plugged in the numbers for the formula of Q=Lm, why did multiply the latent heat by 10^3 ? Isn't it already in kj?

Change Due to Heat

As an object's temperature rises, its dimensions will begin to increase. Different materials have different coefficients of linear expansion (α). The change in length is given by

∆L = L α(∆T).

Similarly, the volume will also increase as the temperature rises. The coefficient of volume expansion is β = 3α.

∆V = V β(∆T).

The above two formulas do not work for gases. They work fine for solids and liquids, but because gases already expand to fill their container, we need a different equation. And for that, we need-

-The Mole (mol) is a measure of how many atoms/molecules/objects there are in a substance. If m is the molecular mass of the substance and M is the mass of the substance in grams, then n=[M/m] is the number of moles.

In a mole there are NA = 6.022 ·1023 [objects/mol]. This is called Avogadro's number.

We can model the behavior of a gas with the Ideal Gas Law:

pV = nRT.

p is the pressure of the gas. [Note: absolute, not gauge pressure].

V is the volume of the gas.

n is the number of moles of gas.

R is the gas constant: R = 8.314 [J/(mol·K)].

T is the temperature (in kelvin) of the gas.

As more and more thermal energy is added to a substance, it will change phase: first from solid to liquid, then from liquid to gas. If thermal energy is removed, it will do the reverse. The specific change over temperatures vary depending on pressure.

It takes thermal energy to change phase. The amount of heat depends on the substance and the mass of the object:

Q = Lm,

where L is a proportionality constant that varies depending on substance and phase type. This heat is put into the substance if it is going to a higher energy phase, and removed if going to a lower energy phase.

Change Due to Heat

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Transcription: Change Due to Heat

Hi. Welcome back to educator.com. Today we’re going to be talking about changes due to heat.0000

We’re already talked about how more thermal energy in a substance causes the molecules of the atoms in the substance to vibrate more and more.0007

As those molecules vibrate more against one another they’re going to be bouncing off each other more often.0014

They’re going to push against each other harder. So as they push harder against each other, they’re going to have more pressure in between them which is going to cause the substance to expand some amount.0020

If it’s a solid substance, we won’t see much expansion. If it’s a liquid substance, we’ll see of that internal pressure in the thermal energy will cause things to push out slightly, a little bit more push in our thing.0030

It’s going to grow a little bit. At very high levels of vibration, very high levels of thermal energy the material will actually change phases as the pressures inside the vibrations inside become so large that they’re able to break being in a solid substance and become able to just move on each other fluidly.0043

Like a in a liquid. Then eventually so much that they’re able to completely pull off of one another and turn into a gas.0059

First idea, linear expansion. When we heat or cool any substance we’re going to get some slight fluctuations in size.0068

The amount of that fluctuation is going to depend on the original length of the object, l. The specific properties of the substance.0074

Some substances will wind up expanding a lot more; some of them won’t expand very much.0080

The change in temperature delta t. Put together we get the equation delta l is equal to l times alpha times our change in temperature.0084

Even the ones that are large fluctuations, the large alphas are still pretty small.0093

All of these alphas are very small. It’s going to basically vary somewhere between 10^-6 and 10^-4 for most substances.0098

Even a really big alpha is still a huge amount of temperature going to have to be changed before we’re actually going to have any really noticeable difference that we’d be able to see with a naked eye.0105

For the most part we aren’t going to actually see this sort of thing. It will have effects on very large structures though, say if you were to build a bridge, you’d want to be able to have that bridge expand and contract without ripping itself apart or smashing into itself.0116

They wind up creating these slots like this so that the bridge can expand into the slots and pull back out without completely ripping itself apart.0129

If they were to build one long many hundreds of meters continuous bridge, when it got very hot it might expand so much that it actually causes itself to pull apart.0140

It’s really important to have those contractions and expansion points otherwise bridges wouldn’t be able to be built.0149

Linear expansion is great and that’s really interesting, but that’s not enough to describe lots of things out there.0156

If we wanted to talk about volume expansion, if we wanted to talk about how much liquid in a cup, what its volume changes to as it gets hotter, it will change.0162

Since its length is changing, but we don’t have length in a liquid. We can’t pour something into some funny shaped container, we can’t pour water into this funny shaped container and say ‘Oh look that has a length of…’0171

It doesn’t make sense. We can’t say…we can’t put a width and length and a depth and be able to easily come up with cubic volume.0184

Instead we just need to be able to talk about volume direct. So it’s very similar, it’s going to be that the change in v is equal to the original volume times beta, the new coefficient for that times the change in temperature.0191

It’s going to turn out that beta isn’t precisely equal to 3alpha but it’s good enough that it’s going to work for all of our purposes and most purposes you’re going to possibly have.0212

Finally we’re ready to talk about gas expansion. Solids, liquids, they both…they push against one another they’re going to expand out just a little bit as they get higher and higher temperatures they’re going to push harder.0221

But a gas, it’s already pushing hard enough and bouncing off of itself. If you put it into a container, it fills out the whole container.0231

It’s not like liquid where it just fills up to the level you poured it in or solid where it just sits there.0237

A gas is going to expand to whatever container size it’s put inside.0242

Since gas is…are already completely filling the container, they’re not going to expand the way we talk about liquid and solids expanding.0246

Instead we’ll talk about how hard they’re pushing. How hard are they pushing against the walls of the containers and how hard they’re pushing against each other.0253

It’s their pressure. In addition we’ll also have to talk about the size of the container, v and the temperature of the gas, t, and how many molecules are in that gas.0261

If we had a box with one atom in it. One single atom in it bouncing around, it’s not going to have that many bounces.0270

If we give it a certain temperature and the same volume, it’s not going to have that many bounces because it’s just one atom.0278

If we were to do the same thing, same size box but now we put in 10 different atoms.0285

If it’s go the same temperature, meaning that each of the atoms is moving at the same rate as this one over here.0295

If we’ve got the same temperature between those two boxes, we’re going to get way more bounces going on in that second box.0301

That means that second box is hitting things more often. Means it’s pushing against stuff more often.0306

It keeps bouncing off repeatedly, that effectively turns into one continuous push when you’ve got millions upon millions upon millions upon millions of molecules doing all of these pushes one after another.0311

That’s what we’re going to feel as pressure. It’s going to really matter how many of these molecules, how many of these atoms do we have doing the bouncing.0322

That’s going to be a really important idea but we haven’t talked about a way to describe how many molecules, how many atoms we have.0329

That’s going to bring up an entirely new idea; it’s going to bring up the mol, which we shorten to m o l.0335

Now we need to talk about mols. Real quick, let’s say you worked in a nail factory and you had a crate of nails.0342

If you needed to know how many nails where in the crate you could count them but that would be a really, really slow process.0349

Say you have a big crate of nails, that’s going to take you hours if not days to be able to count through each of those nails by hand.0354

That’s not a very good way to figure out how many nails you’ve got in the crate.0362

But you might still need to know how many nails you have in the crate if you’re going to do some sort of job in construction or try to sell it somebody; you’re going to need to know how many nails are present.0365

So what you can do instead is you could measure the mass per quantity for a small amount.0372

You might be able to figure out ‘Oh, for every kilogram of nails I have, I’ve got 230 nails in that kilogram.’0379

You’d be able to come up with a conversion ratio of 230 nails per kilograms. Now if you’ve got that you can just dump the whole crate onto a scale.0386

You weigh all of those nails at once and boom; just with a little bit of math you’re able to figure out ‘Oh that’s how many nails I’ve got.’0394

Way, way easier than trying to sit there and do it by hand. Now imagine if you were trying to do something like count molecules.0401

There’s no way we could count it by hand. First off, there’s way too many molecules for us to ever have any hope of counting something by hand.0408

Two, I can’t pick up a molecule. Can you? I can pick up a bunch of molecules at once, but I certainly can’t just pick up one individually and put it over somewhere else.0413

We’ve got no hope of counting molecules but this idea of being able to change mass for quantity is going to work perfectly for us.0421

In general if you know how many objects are some unit mass, you can easily count a substance by using its weight.0429

You figure out what its weight is depending on what local gravity is. In our case we’re on Earth, you’re probably going to be using 9.8.0434

You figure out what its weight is, you then figure out what the mass is from that and using that mass you can boom…if you’ve got a conversion ratio you can easily find out how many of the thing you’ve got.0440

With that idea we can do the exact same thing with molecules. Since every atom or molecule already has an atomic or molecular mass.0449

Where molecular mass is just adding up all of the atomic masses for the number of one’s we’ve got.0459

Depending on what the molecule is. We can connect that to the masses we can measure.0463

We’ve got this idea of molecular mass. You figure out what he molecular mass for your molecule is or the atomic mass if it’s just a single atom.0468

Let little m be the molecular mass. So little m is the molecular mass, we can now also weigh the quantity we have.0476

Say we’ve got some pure amount of lead. We know it’s just lead so we go, we look up the atomic weight for lead and then weigh the quantity of lead we have.0483

We measure it in grams. It’s really important to notice it is grams that we’re using. It’s not kilograms.0493

We normally use kilograms for everything else but we get this idea from Chemistry and because we’re dealing with normally small quantities, we use grams.0499

Measure mols in grams, not in kilograms, in grams. We measure our quantity of lead and then we divide it by what that atomic mass is.0508

If you’re doing it with molecules and you’ve got 1 mol of molecules. Then what you’ve got is 6.022 x 10^23 many molecules.0537

That’s a whole lot of molecules. This number is called Avogadro's number because he did pioneering work in figuring out quantities.0546

He wasn’t actually the person to figure precisely this number but he did a lot of work in figuring out how much of a substance is there and how it connects to other things.0554

Now we’re ready to get back to the idea of expanding a gas. We’re finally ready to relate gas, temperature, and the number of mols, and all these sorts of things.0563

We do this with the ideal gas law, so named because it models an ideal gas. There isn’t such a thing as an actual ideal gas.0570

It’s just a theoretical gas that we use to create a useful formula. It does allow us to really closely model real world gases.0578

It’s not absolutely perfect but for our purposes it’s going to be well within .1% or .01% of what we’re going to have if not even better.0587

It’s going to do great for our problems, but if we were trying to do something really, really careful we’d have to come up with a new way to model it.0593

We’d need more data, but for our purposes it’s really great and it’s a great way for being able to get a close estimation of what’s going to be going on.0601

Now we need to figure out how to read this. This is a lot of letters.0608

P is the pressure of the gas. Note that this absolute pressure, not gauge pressure. If we had air in a tire and we wanted to see what the pressure on that was, we wouldn’t just be able to take a measure on the tire using a gauge.0612

We’d have to figure out what’s the pressure in that tire but then also add what local air pressure is. We’d have to use the absolute pressure.0627

Next up v, is the volume of the gas. Whatever size container it is. Expanding on that tire idea, if would be the volume inside of that tire, how much space the gas is filling out.0634

N is the number of mols of the gas that we have to figure out by knowing what the mass of our quantity is. You’d have to figure out some way to measure the amount of gas you’ve got.0645

If you know how many mols you’ve got you’re ready to keep going. R is just a single number, it’s the gas constant and r is equal to 8.314 joules per mol x kelvin.0655

Interesting thing to note, that also means that the left and the right side wind up equaling joules in the end. Because we’ve got mols x kelvin, so n and t are going to cancel out the mol and kelvin.0666

So we’re only going to be left with joules, so that means that pressure times volume is actually a measure of the energy in the system.0677

The number of molecules you’ve got times the temperature times the special constant is also a measure of how much energy in the system.0685

Which makes sense, a really, really pressurized container has a lot more energy in it than a container that has almost no pressure in it.0692

If you’ve got a balloon, it’s not going to do much if it’s completely deflated and you poke a pin in it. But if you’ve got a fully inflated balloon and you poke a pin in it something’s going to happen.0700

That’s a display of the energy occurring. Not of the energy occurring, I should rephrase that as you seeing how much energy is in that. The energy is already there but you’re now causing it to change in other forms.0708

T is the temperature in kelvin of the gas. Remember we’ve got to have this kelvin otherwise it isn’t going to work.0720

We put all those things together and we’ve got this really useful equation for being able to connect how much the gas will expand and how much the gas will push in on the walls of its container with the volume and the pressure and the number of molecules there are in the special gas constant.0726

One special note on water. Water is really special. Water is maybe almost unique. It might actually be fully unique. It has many, many special properties.0742

Some of those properties are shared with other things but as far as an object that has all of those properties in it. One substance that is all of those properties, it’s really amazing. I don’t know of anything else that has as many extremely unusual properties.0753

Water has a number of these unusual properties and if you’re interested just take a quick internet search on special properties of water or waters properties.0768

You’ll find that there’s a whole bunch of these things and each one of them is incredibly important to the way our world works.0778

First off we already its amazingly high specific heat, previously water has one of the highest specific heats out there.0787

Another thing that’s really strange about it is the fact ice floats in water. You probably haven’t thought about this too much because we’re used to it.0794

Ice has always floated for our lives so we sort of take it for granted. Everything that we’ve learned so far has been the opposite direction.0801

We’ve discussed that every time we have a hot object and you make it hotter it causes it to expand.0810

If you make it colder you cause it to become denser. So a cold object shrinks in is volume while retaining the same mass.0815

That means the colder you make an object the denser you make an object. If you make an object denser, wouldn’t that mean the solid form of it would always sink in the liquid form of it?0822

That is true for most substances, if we did this with a brick of iron and dropped it into a molten container of iron that brick would fall to the bottom before eventually melted by that heated self.0833

For water near the freezing point, at around…I think precisely 4 degrees centigrade…Celsius not centigrade. Celsius, they changed the name a while ago.0843

At 4 degrees Celsius it actually stops contracting and expands to create the crystalline structure we know as ice.0851

At a certain point it begins to line up into ice. Before that it becomes very, very dense. Once it’s at 4 degrees Celsius. But after that point, it starts to get prepared for its transition into ice.0859

Ice is a really specially arranged crystalline structure. And that crystalline structure causes it to have more space in between the molecules than the liquid form does.0871

Since it’s got that more space in its structure it’s going to be less dense than the water and it’s going to actually wind up floating.0880

This is really, really cool and it’s actually one of the things that allows life to exist on Earth.0885

Its possible life would still be able to exist without it but it certainly makes it a lot easier for complex life forms to exist.0892

If it were otherwise, if water were to get really dense when it was frozen then that would mean the coldest part of the water would be on the bottom.0897

That means if you had a lake it would wind up freezing from the bottom up. If it froze from the bottom up that would mean everything in that lake would die.0907

Because it would freeze from the bottom up to the top, you’d have a fully solid lake and fish in there, in plants in there would just wind up dying off.0917

That means that you wind up killing off a huge amount of your aquatic life. Instead the way things occur now ice rises to the top.0924

At the top it winds up actually providing an insulating layer from the cold exterior.0933

Because we’ve got that nice ice on top the fish are able to go down a lower level and they’re able to survive for the winter.0938

They’re able to survive this thing that would otherwise cause all the water to freeze up and kill them off.0945

So because of the fact that ice floats it’s one more reason we can have life on Earth. It’s really cool.0950

Now we’re ready to switch gears and we’re ready to talk about change of phase.0957

So far we’ve talked about expanding within a phase but we haven’t talked about how we’re going to be able to jump from being a solid to a liquid or a liquid to a gas.0960

We want to talk about…so this idea is substances are held together by inner molecule forces. We enough latent energy, enough temperature, the molecules start to bounce around more and more.0968

These inner molecular forces that hold them together can actually be overcome by these being energetic enough to just pop out.0977

If we’re in a solid, we’re in this structure where their fairly rigidly head together. Eventually we’re able to put enough energy in there for instead of being rigidly held together they’ll sort of turn into a slush and they’ll be able to slide on one another.0984

If we put in even more energy they’ll be able to jump out of the slush and fly around. They’ll be able to just fly around whatever container they’re in.0995

If it’s on Earth they’ll just be able to fly through the atmosphere. You’re able to jump out of the liquid with enough energy.1004

As we manage to switch from one phase to the next, we’re clearly going to get some different properties.1010

Solids are different than liquids are different than gases. As more and more heat is added to a substance it changes phase.1014

Starting at a solid and applying heat, it will first melt into a liquid and then it will vaporize into a gas.1021

That’s how we switch forward with more and more energy. If we do the reverse and pull heat energy out, if we cool the substance the process reverses.1028

We start at gas and then it will condense into a liquid and then it freezes into a solid. We first condense into a liquid, it condenses into a liquid and then gas will freeze into a solid. It freezes into a solid.1038

Inner molecular attraction is not the only force holding a substance together. We also have to account for pressure.1053

If we’ve got inner molecular force holding it in, pressure is doing the exact same thing from a different point of view.1059

Its just pushing down. If it’s pushing from all sides, it’s going to help keep whatever it is, it’s going to help keep it together. It’s going to pushing it together all the time.1066

If a substance is immersed in a fluid like our atmosphere the pressure of that fluid is going to keep pushing on the substance and it will back up those inner molecular forces and will help hold the substance together against rising temperatures.1074

While that temperature might be able to defeat the inner molecular force on it, it might not also be able defeat, defeat being sort of an odd word to use here, but it won’t be able to stronger than the force of the pressure and the inner molecular forces.1088

It might be strong enough to overcome one of them but not necessarily both. This means that the phase of substance is determined not by just temperature or pressure; it’s determined by the combination of them.1103

Temperature and pressure. If we want to understand how a substance is going to behave with temperature and pressure we’re going to have to look at both of those together.1112

The specific change over points will vary from substance to substance and to be able to show any given substance we make a phase diagram.1123

A phase diagram will look something like this picture right here. As we have higher and higher temperatures we become things likes gases and liquids.1129

At low temperatures and high pressure, its stays solid but if we drop the pressure, we could become a liquid at the same temperature or even a gas at the same temperature.1139

Notice how easy it is with very low pressures to be able make that switch over from solid to gas. At low pressures it’s easy for the thing to break apart.1148

At higher and higher pressures it takes more and more thermal energy.1158

It’s possible for a solid to transform directly into a gas. If we had a really low pressure we could hop over by just increasing the temperature without ever having to touch liquid.1163

This is a process called sublimation. You’ve probably seen it if you’ve ever played with dry ice or seen dry ice before.1174

It manages to jump directly from being a solid hunk of dry ice into being carbon dioxide, I’m pretty sure its carbon dioxide, but if you’re doing an experiment with it you might want to double check.1180

I should have looked this up beforehand. Anyway, dry ice is able to sublimate directly from its solid from to its gas form.1192

There’s also another special thing to talk about. Right here is the triple point. This is a special point that’s going to occur on each phase diagram at different locations.1199

That’s going to be a unique state where all three forms can exist at once. All three phases will be able to simultaneously exist. You could have solid next to liquid next to gas.1208

Which is a really amazing thing. You couldn’t really imagine something that’s able to be same temperature and have steam and ice and water all floating together happily coexisting.1218

That’s a really, really hard thing to achieve because we have to be at just the right pressure and just the right temperature but it is possible for any substance.1230

Now getting from one phase to the next isn’t just as simple stepping over a line. For a given mass to be able to change phase it has to overcome its heat of transformation.1239

What this means is that to be able to jump from being a solid to a liquid you don’t have to just hit…for water for example we don’t just get to 99 degrees and then 100 degrees and then 101 degrees and then it changes from water, water, gas.1249

Water, water, steam. It doesn’t work like that, instead we get 99 degrees centigrade, 100 degrees centigrade, and then we have to put in a whole bunch more energy before we’re able to get it to jump up to the steam level.1265

There’s a big difference between managing to get up to the line and actually crossing the line on that phase diagram.1275

This is called the heat of transformation. It’s going to vary depending on the mass of the object. More mass means it’ll take more energy to get to jump that line.1283

The amount of energy we take is heat is equal to l x m. Where l is going to be proportionately constant that will vary from substance to substance and what kind of transformation it is.1293

If it’s a solid to a liquid or a liquid gas transformation, it’s going to be different for even the same substance.1304

We’ll have to look up the specific substance we’re talking about and what kind of transformation, but whatever the l is we could figure it out or look it up in a table.1309

If a substance is moving to a higher energy phase like solid to liquid to gas it has to take in q from its environment.1318

If on the other hand it’s going to lower energy phase like gas to liquid to solid it actually releases the energy.1324

When water becomes ice it actually puts out energy into the environment around, which makes a lot of sense because that’s why we have to put it in a freezer, we’re having the environment constantly pump the heat away from that area.1331

That’s what the point of a refrigerator is. We’ve got a heat pump where it takes the heat inside of the box and it puts it somewhere else.1342

Because it’s putting all that heat somewhere else we’ve got a lower environmental temperature which allows things to give off their own temperature and become things like ice for example.1349

Now we’re ready to go on to some examples. First off being able to change the length of temperature.1360

If we’ve got a brass pipe of initial length of 20 meters and its alpha, its coefficient for transformation is 18.7 x 10^-6 per kelvin and it has a temperature rise of 80 kelvins, what would the new length be?1364

Well we’ve got that the change in the length is equal to the original length times alpha times the change in temperature. If we want to know what the change in length is we just toss each of those in.1380

We’ve got 20 meters times 18.7 x 10^-6 times 80 kelvins. We multiply those altogether and we get 0.03 meters. So it manages to gain a positive 0.03 meters.1392

If we want to know what the final length is, we add that to our original length and we’ll get 20.03 meters is our final length.1408

Notice even with the changeover of 80 kelvin, a fairly large amount of temperature from my point of view. It manages to only grow .03; this is a really small amount.1419

The amount of changeover that we’re going to get is really small compared to the temperature difference that we have to have here.1430

We’re either going to have to be talking about absolutely giant lengths or absolutely giant temperatures differences before we’re really going to notice this.1438

This is why we don’t notice it in our daily lives. Although if you wanted to have one experiment or actually a useful thing, if you ever have a stuck jar, one trick you can do is run the lid under really, really hot water.1445

Because metal has a higher coefficient of expansion than the glass so that means the metal on that lid will wind up become a little bit larger and so you can then grab it and twist off and make it a little bit easier to turn and take it off.1456

There of course be careful if you’re going to do this, you are running it under hot water and you don’t want to accidentally burn your hand when you’re taking it off.1469

It will make it slightly easier because you’ll be changing that friction because it will have less pressure holding it to the glass.1475

It will almost always be good enough. Let’s try to figure out how another way, how we could figure out what beta should be.1492

Another way to get to beta. The way that we’ll do this is we’ll start off the cube where all the lengths will undergo linear expansion.1500

What would the new volume of this cube be? Let’s say one of the lengths in our original thing was l. If l is what the original length is, l new, the new l would be equal to l plus the change in l.1508

Well we know what the change in l is; change in l is just the original length times alpha times the change in temperature.1524

Now it’s going to make this problem a lot easier if we set change in temperature equal to 1.1532

Because that change in temperature is just going to wind up showing up throughout the entire problem and it’s just going to be multiplicative fact that keeps stacking and so it’s actually not going to be a problem.1539

We should do this…if were being really, really rigorous we’d want to not make it simpler on ourselves, but we’ll still be able to see the same reasoning so it’s okay in this case to change it just to 1.1549

Now we’ve got at this point that it’s going to be equal. New l is equal to l plus l alpha.1558

What was the old volume of our cube? Our old volume of our cube was equal to l times l times l. Which is equal to l cubed.1566

If we want to know what the new volume is, we’ll that’s going to be v old plus whatever the change in the volume was.1577

Now we want to figure out a way to solve for change in volume. Also notice v old is just the same thing as l cubed.1586

So we’ve got l cubed plus whatever the change in volume wind up being is the way to find out what v new is.1592

If that’s the case, what’s another way to figure out what v new is? We managed to apply that heat so all of our lengths turn into l new. So v news length, the new length…sorry the new volume just has to be based off of the new lengths.1599

The way we find volume is a cube is the side cubed. So l new cubed is equal to v new. We’ve got l new cubed equals l cubed plus change in v.1613

Well l new is the same thing as l plus change in l cube equals l cubed plus change in v.1626

Start working this out, so we’ll have l plus change in l, we can do l plus change in l squared in our head so that becomes l squared...whoops.1634

L squared, what was changed in l, change in l was l alpha, that will make it easier to do. So l plus l alpha cubed, so l plus l alpha cubed. We take off one of those and we’ve now got l plus l alpha times l plus l alpha squared.1645

If we do that in our head we get l squared plus 2 l squared alpha plus l squared alpha squared equals l cubed plus the change in v.1664

We multiply that out some more and we’re going to get l times l squared l cubed plus 2 l cubed alpha plus l cubed alpha squared.1675

If that’s the case, not v cubed. There we are, sorry. So we’ve got a v and v on both sides, so we knock those out and what we’ve really got is 3 v alpha plus 3 v alpha squared plus v alpha cubed equals change in v.1758

Well that’s totally different than the formula we had originally. Which was change in v equals v beta change in temperature, since we’re just dealing with change in temperature is equal to 1; we’ll make that v beta.1780

Since beta we were told was equal to 3 alpha, then we’ve got 3 v alpha. Notice this is totally different than this whole long thing.1797

How is it we can get away with just using v alpha…just using alpha 3 alpha as opposed to 3 alpha plus 3 alpha squared plus 3 alpha cubed.1807

Why? Because alpha is tiny remember, alpha was somewhere between 10^-4 and 10^-6. If its 10^-4 a big alpha, 10^-4 squared is 10^-8.1816

10^-8 is almost certainly going to have no impact practically. It’s going to turn negligible.1832

This right here is going to be really negligible. And this right here is going to be super-duper negligible because of alpha cubed and alpha squared.1838

If we’ve got a gas comprised entirely of O2 with a mass of 1.43 grams and it’s held in a 1 liter container at this pressure and a temperature of this…okay great.1907

If we bring the temperature to a new temperature and we keep the volume fixed what will the pressure increase to?1917

What if we also doubled v while doing it? To do this we’re going to need the ideal gas law.1922

What’s the ideal gas law? It’s the pressure times the volume is equal to the number of mols times the gas constant times temperature.1927

What was the gas constant again? We go, we look it up, we get 8.314 joules per mol per kelvin.1934

To figure out what the number of mols we have is, we’ll have to know that the atomic mass for oxygen. We go we look it up on our periodic table, we get 15.9994 is the mass of oxygen.1946

Remember we’re not going to be dealing with oxygen, we’re dealing with O2. We’re dealing with molecules of oxygen.1962

If the mass for oxygen is 15.9994 the mass for O2 is going to be double that and we’re going to get 31.988 is what we’re going to get for the mass of O2.1968

If that’s the case then our number of mols is the number grams we have of our gas, 1.43 grams divided by the molecular mass, 31.988 and that gives us n is equal to 0.0447 mols.1985

We also need to talk about what’s the temperature. If ti is equal to -.5 degrees Celsius, how do we convert from Celsius to kelvin?2007

Remember we’re not allowed to use Celsius with temperature when we’re dealing with ideal gas law.2020

We can talk about it when we’re just talking about change in tempreture because Celsius and kelvin have the same change size but they have totally different absolute values.2026

If we’re talking about what the full value of it is we’re going to have to switch over to kelvin because the ideal gas law its defiantly using the full value.2036

It needs t, not change in t. So how do we convert over? Temperature in Celsius is equal to the temperature in kelvin minus 273.15.2042

If we want to know what the initial temperature is over here, temperature initial, we can switch it over by adding 273.15 and so we get 272.62 kelvin.2054

What if we want to know what t final is? T final, once again, we had 273.15 and we’re going to get 313.15 kelvin.2065

Interesting thing to note, we didn’t actually have to convert ti. In fact as far as this problems concerned we didn’t ever have to know ti because this formula right here, it doesn’t concern itself with the difference.2078

It only concerns itself with the ending value is. If we had to figure out something from the beginning we would have needed to know what that starting value was, but we didn’t.2090

Everything was given to us, so in the end we actually don’t need to know it because it’s already set what it’s going to wind up being, what it’s going to wind up ending at.2096

At this point we’ve got everything figured out so we can start plugging things in. The equation p v equals n r t.2105

We do the first part of the problem first. What’s the pressure that we’re at? We don’t know what the pressure is, we’re solving for pressure.2113

What’s the volume that we’re at? We’re in a 1 liter container…oh whoops, that’s one more thing we have to solve for.2119

What is 1 liter as a volume? Volume equals 1 liter, we’re not allowed to use liters. We don’t use liters for this, we use cubic meters.2125

One thing we…I’m not sure if you know this already, if we’ve already talked about this. 1 liter…well what’s 1 milliliter? 1 milliliter is the same thing as 1 cubic centimeter.2136

If 1 milliliter is equal to 1 cubic centimeter then 1 liter would a thousand of those. So 10 x10 x 10. So 1 liter is the same thing as .1 meter cubed.2147

So we plug that in and we got pressure times 0.001 cubic meters, remember we can’t use liters for this we have to do it in cubic meters because that’s how everything’s been built.2172

That’s what Pascal is based on. Then we plug in the number of mols we’ve got, 0.0447 mols times the gas constant, 8.314 times the temperature that we end at 313.15.2184

We solve for what the pressure is using a calculator and we get that the pressure wind up being 116,378 Pascals. If we want it to be something a little bit more hand able, that’s approximately same thing as 116.4 Kilopascals.2201

For the second part of this problem, we’re just going to need to also change the volume. So the new volume it’s going to change to is double its original amount, so it’s going to be pressure times 0.002 cubic meters, because it’s doubling to 2 liters now.2219

Is equal to exact same stuff for everything else; we’ve got the same number of mols, same gas constant, same ending temperature.2234

We solve that out for our pressure and get 58,189 Pascals, which is equal to something a little bit more sensible, we’ll talk about it in Kilopascals.2247

Notice changing that volume, since it’s got double the space to bounce around in, it actually winds up being half the amount of pressure.2264

Since they bounce, since they’ve got double the space to bounce around in, you’ve got half the number of bounces occurring on average. So with half the number of bounces occurring on average, you’re going to have half the pressure.2273

We’re ready to do our final example. If we’ve got 2 liters of water on a stove that’s currently at 20 degrees Celsius, how much heat energy cubed do we have to put in that water to bring it to boil?2284

How much more heat energy will we have to put in to boil off all the water?2296

First thing to notice, we don’t have liters as mass. We’ve just got 2 liters of water.2299

If we’re dealing with standard room temperature water and standard pressure, which we are reasonably thing to assume since we’re dealing with a stove.2304

We can convert that since 1 milliliter of water…sorry 1 milliliter of water, standard atmosphere and pressure is the same thing as 1 gram of water.2311

Then a liter of water is the same thing as 1 kilogram. So a liter is a kilogram of water.2323

If a liter is equal to a kilogram then 2 liters is equal to…well they’re not…liters not equal to kilogram, but liter goes to kilogram when it’s in terms of water.2329

So 2 liters goes to 2 kilograms. We’ve got a mass of 2 kilograms of water, which we’re going to need to use to be able to do any of this.2340

Also what’s the specific heat for water? Specific heat for water is equal to 1 calorie per gram per kelvin.2349

If its 1 calorie per gram per kelvin, well that’s the same thing as 4.1868 joules per gram per kelvin.2361

So 4.1868 joules per gram per kelvin, but if we want to do that we can also up convert that to kilograms, 4.1868 kilojoules per kilograms. Kilo both on the top and bottom.2369

Now we’ve got that’s what our specific heat is. That’s how much energy it takes in to be able to raise it 1 degree.2388

If we need to get from 20 degrees Celsius to 100 degrees Celsius, what’s our change have to be?2395

Our change in temperature is going to be equal to 80 kelvin. Since a kelvin and a Celsius are the same thing.2401

We’re going to start boiling once we hit 100 Celsius, but it will take a lot more energy, because we have to then push over. We have to push over all of that extra energy to be able to actually get over that line.2406

To be able to get all of our liquid over the line and into the gas phase. We’re going to need even more energy on top of just getting it to 100 Celsius.2419

Change in temperature equals 80 kelvin. Our formula is the amount of energy that we have to put in to get a given temperature increase for a given substance is the specific heat of that times the mass we’re dealing with times the change in temperature.2428

Our specific heat is 4.1868 kilojoules per kilogram, so its 4.1868 times 10^3 since we’re dealing with kilojoules, times the mass we’re dealing with is 2 kilograms, times the change in temperature, is 80 kelvin.2446

We hit that with a calculator and we get 669,888, way too many significant digits and so that becomes 670 kilojoules.2465

We need 670 kilojoules of energy to be able to move 2 liters of water from room temperature to a boiling temperature.2476

That’s not enough energy to be able to get rid of all it. From there we know how to use the heat of transformation.2485

The heat of transformation to get from liquid into gas, the heat of vaporization for water is 2,257 kilojoules per kilogram.2491

If that’s how much it is we’re going to use heat of transformation is equal to l our coefficient of transformation times the mass we’re dealing with.2502

So l is 2,257 kilojoules, so kilo x 10^3 times 2 kilograms. We get 4,514,000,000 joules which is then the same thing as 4.154 mega joules of energy.2510

4.154 mega joules of energy, that’s more than 6.67 times energy. It takes way, way more energy to get it to actually boil off than it does to get it to the point of where it can just begin to boil.2530

As we add in a little bit of energy a little bit of the water will manage to boil off, but to actually boil off all 2 liters, it takes well more than 6x, almost 7x more energy.2552

We’re not even including the fact we’re going to actually lose some of this heat energy. It doesn’t all perfectly go into heating the water.2563

As it’s a hotter temperature it’s going to be radiating more heat greatly to the area around it.2569

It’s going to take probably even more than 6.67x energy to get that water to actually boil off.2574

That’s a huge difference. That’s the reason why you can bring water to a boil, toss in your spaghetti and come back and not have all the water gone.2579

It’s because it’s actually going to take, if consuming a constant amount of heat going to that pot, it’s actually going to take more than 6x the amount of time to get it to go from starting to boil then all the water gone.2587

If it takes you 10 minutes to bring a given amount of water to boil, you know you’ve got at least an hour before all of that water will have wind up boiling off.2603

So it’s a really interesting thing to think about and that’s specific heat of vaporization is also one of the things that’s so amazing about water. That high specific heat of vaporization means that we can things like sweat happen.2611

Where the amount of heat in our skin gets transferred into the water and since it has such a high vaporization heat, we’re able to transfer lots of energy into it and have it get wicked away into the atmosphere.2621

So we’re able to conveniently, usefully cool ourselves because of this high vaporization heat for water. One of the really cool things that water gives us.2631

Alright, hope you enjoyed that, hope it made a lot sense and we’ll see you later. Bye.2639

Related Books

This book includes a set of features such as Analyzing-Multiple-Concept Problems, Check Your Understanding, Concepts & Calculations, and Concepts at a Glance. This helps the reader to first identify the physics concepts, then associate the appropriate mathematical equations, and finally to work out an algebraic solution.